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Tuesday, December 5, 2006

Susan,I plan on expounding on the following comments in greater detail in my soon to be created blog and I invite you to visit it and comment often when it's ready...I've only read some of your thoughts and feelings regarding standards, testing, NCLB, etc. I admire your passion and courage to take on the establishment and challenge the current 'de-constructive' movement as you might perceive it. I agree wholeheartedly that there is now a testing mania in our country that can potentially do more harm than good. Note that if I weren't a centrist I would have phrased that differently! [I bet you’re reacting negatively to my self-characterization since you may see only 2 camps here in this 'holy war’.] You're probably trying to read between my lines and predict whether I am your friend or foe. I hope we can be friends and enjoy our similarities and differences. I know we will disagree on my next few comments but I need to say them to you with the same emotion and passion you exhibit. Ok, here goes...First, I do not see it as inconsistent that one can have reasonable clear goals for students and still nurture and allow children to develop in their own unique fashion. Since math is my specialty, I will use ladders with its rungs as my metaphor for the acquisition of mathematical skills and concepts. There are different math ladders to climb for the different parts of the whole of mathematics and these ladders are in fact dependent upon each other, but each ladder must be climbed rung by rung. You can;t get to the 5th rung from the bottom if the 2nd, 3rd, and 4th rungs are missing, This is the nature of mathematical knowledge as I see it. Now each child can make it to the next rung in a myriad of ways but she still needs to get there if she wants to continue climbing and eventually move on to more sophisticated math ladders that are even steeper and with more rungs. A child can climb math ladders at her own pace, stopping along the way or even needing to return to lower rungs or starting all over again to regain her strength.This is how I view the need for math standards for bands of grade levels and ultimately for specific math courses at the high school. From my perspective, the mathematical exposure of a student sitting in classroom X in district Y should be approximately the same if that same student moved to classroom Z in district W. To me, this is a no-brainer because it's about content. No one tells me HOW I must teach my Advanced Placement students in Calculus. The College Board does insist however that my students must be exposed to the same core content -- the main ideas and principles of Calculus -- as every other student of AP Calculus. Never once in over 3 decades have I ever felt constrained by this or by the test. My creativity is not restricted, nor do I expect all my students to solve problems the same way. The dialogue in the class is fruitful and thought-provoking. I don't race through the content to finish ahead of everyone else because I understand the nature of how children learn. I believe with conviction that less is truly more when it comes to helping students develop conceptual understanding, but they still need to know their trig identities and the unit circle 'cold'!

In summary, for math at least, the journeys may be different but there are certain destinations one must have in K-8. After that, there can be many different destinations!

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SAT Math Tips

ZERO IS A 'WEIRDO'! (W)hole(E)ven(I)nteger(R)Rational/Real(DO) Cannot Divide by O!BUT Zero is NOT Positive and NOT Negative!

POSITIVE INTEGERS start from 1

PRIMES start from 2 (not 1)

INTEGERS can be NEGative (and zero!) as well as positive

MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.Step 2: Identify the values which are givenFirst term or a(1) = 3Common difference or d = 4Number of terms or position of desired term or n = 100Step 3: Substitute into formula and solvea(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399

Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!

Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!

About Me

Recently retired math educator and Supervisor of Mathematics; 30 years experience as an Advanced Placement Calculus (BC) teacher; Former Author of Math Teachers of New Jersey Annual HS Math Contest; Former K-5 Chair of New Jersey Math Content Standards and Curriculum Frameworks; Former member of Math Item Review Committee for New Jersey High School Proficiency Assessment; Experienced SAT Math Instructor and author of SAT materials; speaker at many regional and national math conferences